\section{极限运算法则}
若$f(x), g(x)$极限都存在，$\lim f(x) = A; \lim g(x) = B$，则：
\begin{gather*}
    \lim \left[f(x) \pm g(x)\right] = \lim f(x) \pm \lim g(x) = A \pm B \\
    \lim \left[f(x) g(x)\right] = \lim f(x) \cdot \lim g(x) = AB \\[5pt]
    \lim \left[\dfrac{f(x)}{g(x)}\right] = \dfrac{\lim f(x)}{\lim g(x)} = \dfrac{A}{B} \qquad (B \neq 0) \\[5pt]
    \lim \left[Cf(x)\right] = C \lim f(x) = CA \\
    \lim f^n(x) = \left[\lim f(x)\right]^n = A^n \\
    \lim \sqrt[n]{f(x)} = \sqrt[n]{\lim f(x)} = \sqrt[n]{A} \\
\end{gather*}

复合函数极限运算法则（换元法）：
$$\text{若} u = \varphi(x), \lim_{x \rightarrow x_0}\varphi(x) = a \text{，则}\lim_{x \rightarrow x_0} f\left[\varphi(x)\right] = \lim_{u \rightarrow a} f(u)$$

若$f(u)$在$u=a$处连续，则$\lim\limits_{x \rightarrow x_0}f[\varphi(x)] = f[\lim\limits_{x \rightarrow x_0} \varphi(x)]$。

\vspace{5ex}
$\lim\limits_{x \rightarrow 0} \dfrac{\sin x}{x} = \lim\limits_{x \rightarrow 0} \dfrac{x}{\sin x} = 1$ \qquad （$\dfrac{0}{0}$未定式）

\vspace{2ex}
$\lim\limits_{x \rightarrow 0} (1 + x)^{\frac{1}{x}} = \lim\limits_{x \rightarrow \infty} (1 + \frac{1}{x})^{x} = e$ \qquad （$1^\infty$未定式）